Dual Elliptic Primes and Applications to Cyclotomy Primality Proving

TL;DR

This paper introduces dual elliptic primes and develops a new primality proving algorithm combining elliptic curves and cyclotomy methods, offering heuristic cubic time complexity and quadratic verification.

Contribution

It extends Galois theory notions to elliptic curves within primality tests, creating a novel algorithm with practical verification benefits.

Findings

Heuristic cubic run time for the new algorithm

Certificates verifiable in quadratic time

Potential for improved primality testing methods

Abstract

Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testing, some interest remains for the practical aspect of state of the art implementable proving algorithms.